No abstract

In this paper we study the simplest deformation on a sequence of orthogonal polynomials, namely, replacing the original (or reference) weight w 0 (x) (supported on R or subsets of R ) by w 0 (x)e −tx . It is a well-known fact that under such a deformation the recurrence coefficients denoted as α n and β n evolve in t according to the Toda equations, giving rise to the time dependent orthogonal polynomials, using Sogo's terminology.If w 0 is the Gaussian density e −x 2 , x ∈ R, or the Gamma density x α e −x , x ∈ R + , α > −1 , then the initial value problem of the Toda equations can be trivially solved. This is because under elementary scaling and translation the orthogonality relations reduce to the original ones. However, if w 0 is the Beta density 1] , α, β > −1, the resulting "time-dependent" Jacobi polynomials will again satisfy a linear second order ode, but no longer in the Sturm-Louville form. This deformation induces an irregular point at infinity in addition to three regular singular points of the hypergeometric equation satisfied by the Jacobi polynomials.We will show that the coefficients of this ode are intimately related to a particular Painlevé V. In particular we show that p 1 (n, t) , where p 1 (n, t) is the coefficient of * Supported in part by NSF Grant DMS-0500892 z n−1 of the monic orthogonal polynomials associated with the "time-dependent" Jacobi weight, satisfies, up to a translation in t, the Jimbo-Miwa σ -form of the same P V ; while a recurrence coefficient α n (t), is up to a translation in t and a linear fractional transformation P V (α 2 /2, −β 2 /2, 2n + 1 + α + β, −1/2). These results are found from combining a pair of non-linear difference equations and a pair of Toda equations.This will in turn allow us to show that a certain Fredholm determinant related to a class of Toeplitz plus Hankel operators has a connection to a Painlevé equation.The case with α = β = −1/2 arose from a certain integrable system and this was brought to our attention by A. P. Veselov.

We prove that the asymptotics of the Fredholm determinant of I − K α , where K α is the integral operator with the sine kernel sin(x−y)π(x−y) on the interval [0, α] is given byThis formula was conjectured by Dyson. The first and second order asymptotics of this formula have already been proved and higher order asymptotics have also been determined. In this paper we solve the remaining outstanding problem of identifying the constant (or third order) term. * tehrhard@mathematik.tu-chemnitz.de. 1 −1 b(x)(2x) k−1 dx, k ≥ 1.

No abstract

The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.